Question

If all the letters of the word 'UDAYPUR' are arranged in all possible permutations and these permutations are listed in dictionary order, then the rank of the word 'UDAYPUR' is

Hint:

While finding dictionary rank, always arrange letters alphabetically first and divide by factorials of repeated letters.

Answer 1

Correct Answer: 1580

Step 1: Arrange the letters in alphabetical order

The letters of the word UDAYPUR are:

U, D, A, Y, P, U, R

Arranging them alphabetically:

A, D, P, R, U, U, Y

Note that the letter U is repeated twice.

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Step 2: Count permutations starting with letters before U

Letters alphabetically before U are:

A, D, P, R (4 letters)

If any of these letters is fixed in the first position, the remaining 6 letters (including two U’s) can be arranged in:

6! / 2! = 360 ways

Total permutations before words starting with U:

4 × 360 = 1440

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Step 3: Fix U as the first letter and count further arrangements

Fix the first letter as U.

Now compare subsequent letters of the word UDAYPUR one by one, counting all valid permutations that come before it.

For the second letter A:

Number of permutations = 5! / 2! = 120

For the third letter D:

Number of permutations = 6

For the fourth letter A:

Number of permutations = 6

For the fifth letter Y:

Number of permutations = 6

For the sixth letter P:

Number of permutations = 1

For the seventh letter R:

Number of permutations = 1

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Step 4: Add all the counts

Total rank =

1440 + 120 + 6 + 6 + 6 + 1 + 1

= 1580

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Final Answer:

The rank of the word UDAYPUR is
1580